Influence of the Baer–Kaplansky Theorem on the Development of the Theory of Groups, Rings, and Modules

The review presents an analysis of results the Abelian group theory, as well rings and modules, which concern definability algebraic structures by their endomorphism related structures. In systematization results, greatest attention is paid to torsion-free groups, are particular interest due presence non-isomorphic direct decompositions in this class. This significantly expands understanding general, including modern, trends development algebra context Baer–Kaplansky theorem. reflection properties objects a certain class natural structural connection, study separate investigation direction. A striking introduction topic was theorem for torsion dates back middle last century states that any isomorphism two groups from inevitably induced some themselves. Of course, it follows if have isomorphic rings, then they isomorphic. profound result inspired mathematicians obtain same form concerning other classes objects. But even theory itself, were discovered analogue valid. Despite fundamental difference between definitions completely decomposable sums rank-one considered, finite order cyclic there one very important common characteristic these classes: into indecomposable summands determined uniquely up isomorphism. property not possessed whose focus our attention.